The Temporal Scale Invariance Hypothesis as a Bridge between Quantum Mechanics and Gravitation

Technical Summary: The Temporal Scale Invariance Hypothesis as a Bridge between Quantum Mechanics and Gravitation

Problem Statement
The paper addresses the fundamental lack of unification between Quantum Mechanics (QM) and General Relativity (GR). While standard approaches (e.g., string theory, loop quantum gravity) rely on complex mathematical frameworks and new entities, this work proposes an alternative perspective. It seeks to resolve the "pilot wave" mystery of the de Broglie–Bohm theory—specifically the physical nature of the wave and its existence in abstract configuration space—and to establish a direct link between quantum phenomena and gravity without abandoning classical determinism.

Methodology and Core Postulates
The author proposes the Temporal Scale Invariance Hypothesis, based on four fundamental postulates:

1. Unified Laws: The micro- and macro-worlds are governed by the same physical laws, differing only by the temporal distance of observation.
2. Time as Scale: The difference between particle and planetary behavior is defined by a time scale coefficient, $\gamma = \tau/t$, where $\tau$ is the system's characteristic time and $t$ is the observer's. $\gamma$ is derived geometrically ($\gamma \approx R_{obs}/r_{sys}$) or via mass ($\gamma \propto m^{-4/3}$).
3. Definite Trajectory: Elementary particles are localized material objects with definite trajectories in their proper time. Probability clouds are not intrinsic properties of the particle but observable manifestations of a "temporal wake."
4. Temporal Wake as a Physical Field: The motion of a massive particle in its ultrafast proper time generates a physical disturbance in 3D space called a "temporal wake." This wake manifests as interference, entanglement, and gravity (the cumulative wake of a massive body).

Key Contributions and Theoretical Framework

• Three Laws of Scale Invariance:

  • First Law: The perturbation function enhancing gravitational interaction at the micro-level follows a power law, $f(\gamma) = \gamma^\alpha$. Calibration on the hydrogen atom yields an empirical exponent $\alpha_{base} \approx 0.915$.
  • Second Law: The time scale is inversely related to mass, $\gamma \propto m^{-4/3}$, implying that more massive particles have slower internal time.
  • Third Law: The connection between geometric and mass-based approaches is established via density, $\gamma \propto (\rho/m)^{1/3}$, implying particle density decreases as mass increases ($\rho \propto m^{-3}$).

• Coupled Field Equations:
  The paper proposes a system of coupled field equations for $N$ massive particles. Each particle generates a field $S_i(x,t)$ (the temporal wake) obeying a hybrid equation involving a d'Alembertian and a mass term dependent on $\gamma$.
  $$\Box S_i - \frac{1}{\lambda_i^2} \left(1 - \frac{1}{\gamma_i}\right) S_i = J_i + \alpha_{eff} \sum_{j \neq i} S_j \cdot \delta(x - x_i)$$
  The coupling between fields is mediated by an effective exponent, $\alpha_{eff}$, which is not a constant but a function of time scale ($\gamma$), system symmetry ($\mu$), and density ($\rho$).

• Phase Transition Mechanism:
  The model describes a natural phase transition between three regimes based on $\alpha_{eff}$:

  • Atomic Regime ($\rho \ll \rho_{crit}, \gamma \gg 1$): Fields overlap weakly; coupling is active ($\alpha_{eff} \approx \alpha_{base}$), producing quantum potentials and interference.
  • Nuclear Regime ($\rho \approx \rho_{crit}, \gamma \gg 1$): Fields fully overlap and become identical ($\alpha_{eff} \to 0$), explaining confinement as a coherent state rather than a strong force property.
  • Classical Regime ($\gamma \to 1$): Coupling switches off ($\alpha_{eff} \to 0$), recovering Newtonian gravity.

Results and Quantitative Verification

• Toy Model Verification: A perturbation model was tested against experimental binding energies for systems ranging from the hydrogen atom to macroscopic bodies (Sun-Earth).

  • The model achieved order-of-magnitude agreement for atomic systems.
  • It showed good agreement for light nuclei (Deuteron, Alpha particle) with deviations of 6–7%.
  • It achieved 0% deviation for the Sun-Earth system.
  • The initial discrepancy of ~40 orders of magnitude between gravitational and nuclear/electromagnetic interactions was reduced to a coefficient of order unity.

• Fine-Structure Constant ($\alpha_{fs}$): The paper derives a formula expressing $\alpha_{fs}$ through gravitational parameters and the time scale: $\alpha_{fs} \approx \frac{G m_1 m_2}{\hbar c} (\gamma_1 \gamma_2)^\alpha$. This formula reproduced the order of magnitude of $\alpha_{fs}$ for seven different systems, though deviations were noted for nuclear systems and those with specific symmetries.

• Weak Interaction: The model systematically underestimated the weak coupling constant by a factor of 67–1300, which the author attributes to the need for modifications regarding short-range forces and boson masses.

• Limitations Identified:

  • The current formula for $\alpha_{eff}$ lacks a symmetry correction term ($\Delta \alpha_{sym}$), leading to overestimation of binding energy in muonium and protonium.
  • The exponent $\alpha_{base}$ remains an empirical parameter not derived from first principles.
  • The system requires numerical solutions for full verification, particularly in the transition region where $r \sim \lambda$.

Significance and Claims
The paper claims to offer a phenomenological justification for unifying QM and GR by interpreting quantum effects as manifestations of a single physical field (the temporal wake) observed at different time scales.

• Unification: It posits that gravity, quantum interference, and entanglement are different regimes of the same field dynamics.
• Determinism: It restores a definite trajectory for particles, resolving the configuration space issue of pilot-wave theory by placing the "wave" in 3D space.
• Confinement: It reinterprets quark confinement as a phase transition where individual wakes collapse into a unified field at nuclear density.
• Falsifiability: The work emphasizes its status as a hypothesis subject to experimental refutation. It formulates eleven specific predictions, including:
  • The disappearance of interference at a temporal resolution of $\sim 10^{-23}$ s.
  • The breakdown of entanglement correlations at distances $D > \lambda$.
  • A phase transition at nuclear density observable in heavy-ion collisions.
  • A violation of the inverse-square law at small distances.

The author concludes that while the mathematical ansatz requires further refinement and numerical solution, the hypothesis provides a testable framework that bridges the gap between the quantum and classical worlds through the concept of temporal scale invariance.